Du, Hailiang (2009) Combining statistical methods with dynamical insight to improve nonlinear estimation. PhD thesis, The London School of Economics and Political Science (LSE).

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Abstract
Physical processes such as the weather are usually modelled using nonlinear dynamical systems. Statistical methods are found to be difficult to draw the dynamical information from the observations of nonlinear dynamics. This thesis is focusing on combining statistical methods with dynamical insight to improve the nonlinear estimate of the initial states, parameters and future states. In the perfect model scenario (PMS), method based on the Indistinguishable States theory is introduced to produce initial conditions that are consistent with both observations and model dynamics. Our methods are demonstrated to outperform the variational method, Fourdimensional Variational Assimilation, and the sequential method, Ensemble Kalman Filter. Problem of parameter estimation of deterministic nonlinear models is considered within the perfect model scenario where the mathematical structure of the model equations are correct, but the true parameter values are unknown. Traditional methods like least squares are known to be not optimal as it base on the wrong assumption that the distribution of forecast error is Gaussian IID. We introduce two approaches to address the shortcomings of traditional methods. The first approach forms the cost function based on probabilistic forecasting; the second approach focuses on the geometric properties of trajectories in short term while noting the global behaviour of the model in the long term. Both methods are tested on a variety of nonlinear models, the true parameter values are well identified. Outside perfect model scenario, to estimate the current state of the model one need to account the uncertainty from both observatiOnal noise and model inadequacy. Methods assuming the model is perfect are either inapplicable or unable to produce the optimal results. It is almost certain that no trajectory of the model is consistent with an infinite series of observations. There are pseudoorbits, however, that are consistent with observations and these can be used to estimate the model states. Applying the Indistinguishable States Gradient Descent algorithm with certain stopping criteria is introduced to find relevant pseudoorbits. The difference between Weakly Constraint Fourdimensional Variational Assimilation (WC4DVAR) method and Indistinguishable States Gradient Descent method is discussed. By testing on two systemmodel pairs, our method is shown to produce more consistent results than the WC4DVAR method. Ensemble formed from the pseudoorbit generated by Indistinguishable States Gradient Descent method is shown to outperform the Inverse Noise ensemble in estimating the current states. Outside perfect model scenario, we demonstrate that forecast with relevant adjustment can produce better forecast than ignoring the existence of model error and using the model directly to make forecasts. Measurement based on probabilistic forecast skill is suggested to measure the predictability outside PMS.
Item Type:  Thesis (PhD) 

Additional Information:  © 2002 Hailiang Du 
Library of Congress subject classification:  H Social Sciences > H Social Sciences (General) H Social Sciences > HA Statistics 
Sets:  Departments > Statistics Collections > LSE History of Thought theses 
Supervisor:  Smith, Leonard 
URI:  http://etheses.lse.ac.uk/id/eprint/66 
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